What do the notations $\sim$ and $\Delta (A) $ mean in the paper "Fairness Through Awareness"?
The $\sim$ symbol means that a random variable is drawn from the given distribution, i.e. if I were to say $X$ has a Standard Normal distribution I would write $X \sim \text{Normal}(0,1)$. They write two explicit expectations here because $a$ is a random variable with distribution $\mu_x$ but $X$ is also a random variable with distribution $V$. I believe you are right that $V$ would be analogous to a set of features. So we are saying that $X$ is a random variable over the distribution over the features, or individuals in this context.
As for the $\Delta(A)$, I have never seen this notation used before -- I am not sure if it is standard notation. I Googled to see if it was something that I just hadn't seen before, but there was no such answer. However, from the context of the paper they define $M: V \rightarrow \Delta (A)$ to be mappings from individuals to probability distributions over $A$, so I guess that $\Delta (A) $ is probability distributions over $A$.